17 - A. Yarosh
Retract -1. Bh2-g1 Kd8-e8 -2. Bd6-h2 Kc8-d8 -3. Bc5-d6
(if white retracted Bd4-e5-h2, black can now retract Kb8-c8!, forcing
white to retract Nc5-a6) Rd8-f8 -4. Ng6xQh8! (-4. Ba7-c5?
OOO!) and now black can't retract queenside castling, because the queen
would be trapped in the h8-corner. Finish now with -5. Ba7-c5
-6. Bb8-a7 -7. a7xBb8=B, and play forward 1. a7xb8=Q/R#
However, this is cooked: -4. Ba7-c5! OOO -5. Ng6-h8 Rf8-g8 -6.
Nf4-g6 Rg8-f8 -7. Ne6-f4 and play forward 1. Na6xc7#.
Black can't uncapture anything on his 5th move: The only piece which
could've reached g8 is a knight, and this requires too many captures:
a7xb8=N, and cxbxaxb8=B. The bishops were captured on their home squares.
Because black retracted castling, the king and rook are immovable.

31 - V. Peretyatko
1. Bc4#? But black's last move could've only been e4xf3. But the only
missing white piece is the black-squared bishop (Bh8 is clearly promoted).
So black has the move.
0. .. Rb5/Bb5/Kb5/a3 1. Nc4/Nd5/Qb1/Bc4#

32 - M. Kormilshev
Add a black rook on a1. All black captures were done by the two pawns
on h2 and h3, so black has no possible last move. 1. ... Rc1! and black
doesn't lose.

65 - V. Peretyatko
White has three promoted pieces. One is the g-pawn, which promoted after
g6xh7-h8=R, one is the c-pawn which promoted after c5xBd6-d8=R, and
the last is the a-pawn which promoted on a8 to a bishop. The black a-pawn
was captured on its own file.
Because white captured the bishop on d6, black can't retract d6-d5 too
soon. Because the white bishop had to leave a8, black can't retract
b7-b6 too soon. Because the bishop from f8 was captured on d6, black
can't retract e7-e6 too soon. So until something is uncaptured, black
only has king-moves. The last moves must've been -1. Nf3-e1
Kf1-e2 -2. Rg2-g1 Ke2-f1 -3. Nf2-d1 Kf1-e2 -4. Ne1-f3 Ke2-f1 -5. Rf3-f4
Kf1-e2, and the whole position can get released.
So the mate is legal.

66 - A. Kornilov & A. Frolkin
The position unlocks after black can retract h6-h5, and white can uncapture
h5xg6. For this, a black piece is needed on g8. A white knight can uncapture
this piece on g7. So black needs to uncapture a white knight. The white
captures are also g2xf3, h5xg6, dxexf and Nxg7. So the black a-pawn
promoted on b1, after capturing once on the b-line. So the black captures
are axb, bxc, cxd and exd. So black must uncapture a knight with his
a-pawn, after which this pawn promotes. So black must unpromote Ng1
on b1.
The last moves were: -1. g6-g7 h4-h3 -2. a6-a7 Nh3-g1 -3. a5-a6
Nf4-h3 -4. a4-a5 Nd3-f4 -5. a3-a4 Nb4-d3 -6. a2-a3 Nc2-b4 -7. b6-b7
Na3-c2 -8. b5-b6 Nb1-a3 -9. b4-b5 b2-b1=N -10. b3-b4 a3xNb2 -11. Nd3-b2
a4-a3 -12. Nf4-d3 a5-a4 -13. Nh5-f4 a6-a5 -14. Ng7-h5 a7-a6 -15. Ne8xRg7
Rg8-g7 -16. Qh8-h7 h5-h4 -17. Kh7-h6 h6-h5 -18. h5xNg6 and
the whole position unlocks.
But this is cooked: Retroplay could be too: b4-b7; (b7xBc6);
c2-c7; (c7xBd6); (d6-d5); a4xSb3-b1=B-e6.

Shakhmatnaya Kompozitsiya 32

82 - V. Peretyatko
Black still has 8 pawns, so Bg1 is the original bishop from f8. So the
move-order is: g7-g6; Bf8-g1; Bc1-h2; g2-g3. So g6 was already there
when g2-g3 was played. So Bh7 is a promoted white pawn. It must be the
white b-pawn, which played b2xc3xd4xe5xf6xg7-g8=B. Together with the
bishop on c8, these are all the missing black pieces. So white's last
move wasn't a capture. The only way the position can be explained is
if the last moves were -1. Nc1-d3 Ka1xNb1 -2. Nc3-b1 Kb1-a1
-3. Nd5-c3 Ka1-b1 -4. Nf6-d5. The white bishop on f1 was captured
on its homesquare, and on b6, black captured the missing rook. So the
position is indeed legal.

85 - V. Liskovets
Suppose white may still castle. The white pawns captured seven times:
b2xa3, c2xb3xa4, dxcxbxa7, and fxg. None of these captures were on the
d-line, so either the black d-pawn promoted, or it's one of the pawns
on the g- or h-file. In the first case, because white may still castle,
black had to capture dxc before the promotion. So the black captures
are gxh, fxgxh, dxc, and exfxg. None of these captures is on the e-file,
so the e-pawn had to promote to get captured. So black may not castle
in this case. In the other case, black captured six times with the pawns
currently on the g- and h-file. This would imply the black e-pawn promoted
without capture, which isn't possible, since white may still castle.
So castlings are mutually exclusive.
h#2 with 1. Rd8 OO 2. Rf8 Re1# or 1. hg4 Ke2 (OO?) 2. OOO Rc1#.

87 - V. Soloviev
g2-g3 was played after Bg3-h2, and b2-b3 was played after B-a2. So one
of the bishops b1 or h2 is promoted, and the original bishop was captured.
Black's only capture was g7xf6, so the bishop on h2 is a promoted one.
So a2 promoted on a black square, and he needs three captures for that.
Together with h2xg3xf4 that's all missing black pieces.
First bring the white king to e1, the white queen to d1, the white bishop
to a2, the white rook to b5, and the white knights to b1 and g1. Then,
after taking the black bishop back to f8, uncapture g7xf6. If black
may still castle in the diagram position, he now only has waiting moves
with his pawns. The fastest way to a white uncapture is: -1.
.. g7xf6 -2. Bb2-f6 c5-c4 -3. Bc1-b2 d6-d5 -4. b2-b3 c6-c5 -5. Bd5-a2
c7-c6 -6. Bg2-d5 h5-h4 -7. Bf1-g2 h6-h5 -8. g2-g3 h7-h6 -9. Bg3-h2 a6-a5
-10. Bh4-g3?? -11. g3xf4.. If black may still castle, he has
no move on the question marks.
So black may no longer castle. Mate in 3 with 1. Qb4 (2. Qe7) c3 2.
dc3 (3. Qe7) ab4/Kf8 3. Ra8/Rc8#, or 1. Qa5 c3 2. Qc3 ~ 3. Ra8#.

104 - A. Kislyak
White captures are gxPh, axNb, and Bc8 on his homesquare. Black captures
are c7xb6xa5 and the f-pawn on its own file. The position unlocks when
black uncaptures b6xNa5. For this, the white king must leave the 8th
rank.
Last moves were: -1. Kb8-c8 Kd8-e8. Now the black king
makes waitingmoves between d8 and e8, while the white knights go to
c8 and h8. The white pawn on h7 goes to h2, the knight on h8 unpromotes
and then the retroplay is: Kd8-e8 g3xPh4 Ke8-d8 Kc7-b8 h5-h4
Kb6-c7. Now black can uncapture b6xNa5, and the whole position
unlocks.

105 - Y. Lebedev
The last moves were: -1. .. f7xNe6 -2. Nd8-e6 e5xNf4 -3. d7-d8=N
d6xRe5 -4. c6xQd7 Qd8-d7. White now unpromotes the uncaptured
knight from f4 on a8, and retracts this pawn to a2. Black uncaptures
a3xRb2. Both uncaptured rooks from b2 and e5 unpromote on g8, black
brings back his bishop to f8, and uncaptures g7xPf6, and the whole position
unlocks.
So the game could've started with 1. c2-c3.

106 - V. Peretyatko
Visual black captures are fxe, cxdxexfxg2, and b3xc2. Black needs to
uncapture a white piece with his last move, or white is retropat. This
accounts for all missing white pieces. So white can't retract c2-c3
until one of the queens is unpromoted, and he can't retract d2-d3 until
Bc1 (and Ra1) are back on their homesquare again.
The last moves were: -1. .. Rg7xNh7 -2. Nf8-h7 Be8-f7 -3. f7-f8=N
Qe5-f6 -4. f6-f7 Rf4-f5 -5. f5-f6 Kf6-g5 -6. h4-h5 and the
position unlocks.
So the mate is legal.